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Extremum Seeking is Stable for Scalar Maps that are Strictly but Not Strongly Convex

Patrick McNamee, Miroslav Krstić, Zahra Nili Ahmadabadi

Abstract

For a map that is strictly but not strongly convex, model-based gradient extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at exponential convergence. Interestingly, perturbation-based model-free extremum seeking has a negative Jacobian, in the average, meaning that its (practical) convergence is exponential, even though the map's Hessian is zero at the extremum. While these observations for the gradient algorithm are not trivial, we focus in this paper on an even more nontrivial study of the same phenomenon for Newton-based extremum seeking control (NESC). NESC is a second-order method which corrects for the unknown Hessian of the unknown map, not only in order to speed up parameter convergence, but also (1) to make the convergence rate user-assignable in spite of the unknown Hessian, and (2) to equalize the convergence rates in different directions for multivariable maps. Previous NESC work established stability only for maps whose Hessians are strictly positive definite everywhere, so the Hessian is invertible everywhere. For a scalar map, we establish the rather unexpected property that, even when the map behind is strictly convex but not strongly convex, i.e., when the Hessian may be zero, NESC guarantees practical asymptotic stability, semiglobally. While a model-based Newton-based algorithm would run into non-invertibility of the Hessian, the perturbation-based NESC, surprisingly, avoids this challenge by leveraging the fact that the average of the perturbation-based Hessian estimate is always positive, even though the actual Hessian may be zero.

Extremum Seeking is Stable for Scalar Maps that are Strictly but Not Strongly Convex

Abstract

For a map that is strictly but not strongly convex, model-based gradient extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at exponential convergence. Interestingly, perturbation-based model-free extremum seeking has a negative Jacobian, in the average, meaning that its (practical) convergence is exponential, even though the map's Hessian is zero at the extremum. While these observations for the gradient algorithm are not trivial, we focus in this paper on an even more nontrivial study of the same phenomenon for Newton-based extremum seeking control (NESC). NESC is a second-order method which corrects for the unknown Hessian of the unknown map, not only in order to speed up parameter convergence, but also (1) to make the convergence rate user-assignable in spite of the unknown Hessian, and (2) to equalize the convergence rates in different directions for multivariable maps. Previous NESC work established stability only for maps whose Hessians are strictly positive definite everywhere, so the Hessian is invertible everywhere. For a scalar map, we establish the rather unexpected property that, even when the map behind is strictly convex but not strongly convex, i.e., when the Hessian may be zero, NESC guarantees practical asymptotic stability, semiglobally. While a model-based Newton-based algorithm would run into non-invertibility of the Hessian, the perturbation-based NESC, surprisingly, avoids this challenge by leveraging the fact that the average of the perturbation-based Hessian estimate is always positive, even though the actual Hessian may be zero.
Paper Structure (11 sections, 7 theorems, 34 equations, 5 figures)

This paper contains 11 sections, 7 theorems, 34 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Perturbations Remove Strong Convexity Restriction in Gradient ESC
  3. Newton ESC and Its Average
  4. Problem Formulation and Assumptions
  5. Motivating Example
  6. Main Results
  7. Averaged Derivative Estimates
  8. NESC Practical Stability
  9. Conclusions
  10. Practical Stability
  11. Convexity Lemmas

Key Result

Theorem 1

Let the system described in equations eq:parameter-estimate-ode-eq:invhessian-estimate-ode be acting on a static, scalar map $J$ satisfying Assumptions asmp:twice-differentiable-asmp:global-minimizer. Then in the error variables where $\theta_{*}\in\mathbb{R}$ is a unique point, the NESC system is sGPUAS to the origin. Furthermore, the point $(\hat{\theta},\hat{\Gamma})=(\theta_{*}, 1/H(\theta_{*

Figures (5)

  • Figure 1: Scalar GESC Algorithm
  • Figure 2: Scalar NESC Algorithm
  • Figure 3: Numerical simulated trajectories of the NESC using Eq. \ref{['eq:example-cost-function']}. Simulation parameters were $a=0.5$, $\omega = 10$, and $k=\omega_l= 0.001$.
  • Figure 4: Numerical simulated parameter estimates of the NESC using Eq. \ref{['eq:example-cost-function']}. Simulation parameters were $a=0.5$, $\omega = 10$, and $k=\omega_l= 0.001$. Initial conditions of $(\hat{\theta},\hat{{\Gamma}})$ were (1,5/6), (1,5/3), (1,5/6) and (1,5/3) for the blue, orange, red and green lines respectively. After 5,000 seconds, all the parameter estimates increase their convergence rate and appear to converge exponentially to a neighborhood of $\theta_{*}$, with $\Vert \hat{\theta}(t) - \theta_{*}\Vert <$10e-3 for $t > 8,000$ seconds.
  • Figure 5: The auxiliary function $h$ constructed from support lines and convex line segments for an example $J$.

Theorems & Definitions (14)

  • Theorem 1: The NESC system is sGPUAS
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • proof : Proof of Theorem \ref{['thm:nesc-sgpuas']}
  • Definition 1: ref:labar-2019
  • Definition 2
  • ...and 4 more